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Those messages shown at the bottom of the admin page were sent through Components - Messaging.

You seem to have found a bug. Only reading the message marks it as read. The available selections to trash or marking a message as read or unread, do not have any effect. Please report this behaviour in the Issue Tracker at https://issues.joomla.org/

When I attempted the operations, Joomla displayed only normal messages like "0 messages trashed" or "0 messages marked as unread" but no errors. The javascript console showed the following warnings, which may explain the issue: